Is sec(x)-tan(x)(sin(x))= 1/(sec(x)) ?

The answer to whether sec(x)-tan(x)(sin(x))= 1/(sec(x)) is True

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prove sec(x)-tan(x)(sin(x))= 1/(sec(x))

prove

sec (x) - tan (x) (sin (x)) = \frac{1}{sec ( x )}

True

Solution steps

sec (x) - tan (x) sin (x) = \frac{1}{sec ( x )}

Manipulating left side

sec (x) - tan (x) sin (x)

Express with sin, cos

sec (x) - sin (x) tan (x)

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

= \frac{1}{cos ( x )} - sin (x) tan (x)

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

= \frac{1}{cos ( x )} - sin (x) \frac{sin ( x )}{cos ( x )}

Simplify

\frac{1}{cos ( x )} - sin (x) \frac{sin ( x )}{cos ( x )} : \frac{1 - sin ^{2} ( x )}{cos ( x )}

\frac{1}{cos ( x )} - sin (x) \frac{sin ( x )}{cos ( x )}

sin (x) \frac{sin ( x )}{cos ( x )} = \frac{sin ^{2} ( x )}{cos ( x )}

sin (x) \frac{sin ( x )}{cos ( x )}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{sin ( x ) sin ( x )}{cos ( x )}

sin (x) sin (x) = sin^{2} (x)

sin (x) sin (x)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

sin (x) sin (x) = sin^{1 + 1} (x)

= sin^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= sin^{2} (x)

= \frac{sin ^{2} ( x )}{cos ( x )}

= \frac{1}{cos ( x )} - \frac{sin ^{2} ( x )}{cos ( x )}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 - sin ^{2} ( x )}{cos ( x )}

= \frac{1 - sin ^{2} ( x )}{cos ( x )}

= \frac{1 - sin ^{2} ( x )}{cos ( x )}

Rewrite using trig identities

\frac{1 - sin ^{2} ( x )}{cos ( x )}

Use the Pythagorean identity:

1 = cos^{2} (x) + sin^{2} (x)

1 - sin^{2} (x) = cos^{2} (x)

= \frac{cos ^{2} ( x )}{cos ( x )}

Cancel the common factor:

cos (x)

= cos (x)

= cos (x)

Use the basic trigonometric identity:

cos (x) = \frac{1}{sec ( x )}

= \frac{1}{sec ( x )}

We showed that the two sides could take the same form

\Rightarrow True

p ro v e \frac{1 - tan ^{4} ( x )}{sec ^{2} ( x )} = 1 - tan^{2} (x)

p ro v e 3 - sec^{2} (x) = 2 - tan^{2} (x)

p ro v e (cot (x) + tan (x))^{2} = csc^{2} (x) sec^{2} (x)

p ro v e cos (3 x) = 4 cos (3 x) - 3 cos (x)

p ro v e sec^{2} (a) + csc^{2} (a) = sec^{2} (a) csc^{2} (a)

Is sec(x)-tan(x)(sin(x))= 1/(sec(x)) ?
The answer to whether sec(x)-tan(x)(sin(x))= 1/(sec(x)) is True